Numerical methods for the bidimensional Maxwell-Bloch equations in nonlinear crystals

Two numerical schemes are developed for solutions of the bidimensional Maxwell-Bloch equations in nonlinear optical crystals. The Maxwell-Bloch model was recently extended [C. Besse, B. Bidegaray, A. Bourgeade, P. Degond, O. Saut, A Maxwell-Bloch model with discrete symmetries for wave propagation in nonlinear crystals: all application to KDP, M2AN Math. Model. Numer. Anal. 38 (2) (2004) 321-344] to treat anisotropic materials like nonlinear crystals. This semi-classical model seems to be adequate to describe the wave-matter interaction of ultrashort pulses in nonlinear crystals [A. Bourgeade, O. Saut, Comparison between the Maxwell-Bloch and two nonlinear maxwell models for ultrashort pulses propagation in nonlinear crystals, submitted (2004)] as it is closer to the physics than most macroscopic models. A bidimensional finite-difference-time-domain scheme, adapted from Yee [IEEE Trans. Antennas Propag. AP-14 (1966) 302-307], was already developed in [O. Saut, Bidimensional study of the Maxwell-Bloch model in it nonlinear crystal, submitted (2004)]. This scheme yields very expensive computations. In this paper, we present two numerical schemes much more efficient with their relative advantages and drawbacks. (c) 2005 Elsevier Inc. All rights reserved.